Integrand size = 17, antiderivative size = 78 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )}{3^{5/6}}-\frac {\log \left (-1+x^3\right )}{6 \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3} x-\sqrt [3]{2+x^3}\right )}{2 \sqrt [3]{3}} \] Output:
-1/3*arctan(1/3*(1+2*3^(1/3)*x/(x^3+2)^(1/3))*3^(1/2))*3^(1/6)-1/18*ln(x^3 -1)*3^(2/3)+1/6*ln(3^(1/3)*x-(x^3+2)^(1/3))*3^(2/3)
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx=\frac {-6 \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2+x^3}}\right )+\sqrt {3} \left (2 \log \left (-3 x+3^{2/3} \sqrt [3]{2+x^3}\right )-\log \left (3 x^2+3^{2/3} x \sqrt [3]{2+x^3}+\sqrt [3]{3} \left (2+x^3\right )^{2/3}\right )\right )}{6\ 3^{5/6}} \] Input:
Integrate[1/((-1 + x^3)*(2 + x^3)^(1/3)),x]
Output:
(-6*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(2 + x^3)^(1/3))] + Sqrt[3]*(2*Log[- 3*x + 3^(2/3)*(2 + x^3)^(1/3)] - Log[3*x^2 + 3^(2/3)*x*(2 + x^3)^(1/3) + 3 ^(1/3)*(2 + x^3)^(2/3)]))/(6*3^(5/6))
Time = 0.17 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {901}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (x^3-1\right ) \sqrt [3]{x^3+2}} \, dx\) |
\(\Big \downarrow \) 901 |
\(\displaystyle -\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{3^{5/6}}-\frac {\log \left (x^3-1\right )}{6 \sqrt [3]{3}}+\frac {\log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+2}\right )}{2 \sqrt [3]{3}}\) |
Input:
Int[1/((-1 + x^3)*(2 + x^3)^(1/3)),x]
Output:
-(ArcTan[(1 + (2*3^(1/3)*x)/(2 + x^3)^(1/3))/Sqrt[3]]/3^(5/6)) - Log[-1 + x^3]/(6*3^(1/3)) + Log[3^(1/3)*x - (2 + x^3)^(1/3)]/(2*3^(1/3))
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Time = 2.79 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.19
method | result | size |
pseudoelliptic | \(\frac {3^{\frac {2}{3}} \ln \left (\frac {-3^{\frac {1}{3}} x +\left (x^{3}+2\right )^{\frac {1}{3}}}{x}\right )}{9}-\frac {3^{\frac {2}{3}} \ln \left (\frac {3^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} \left (x^{3}+2\right )^{\frac {1}{3}} x +\left (x^{3}+2\right )^{\frac {2}{3}}}{x^{2}}\right )}{18}+\frac {3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{\frac {2}{3}} \left (x^{3}+2\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right )}{3}\) | \(93\) |
trager | \(\text {Expression too large to display}\) | \(902\) |
Input:
int(1/(x^3-1)/(x^3+2)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/9*3^(2/3)*ln((-3^(1/3)*x+(x^3+2)^(1/3))/x)-1/18*3^(2/3)*ln((3^(2/3)*x^2+ 3^(1/3)*(x^3+2)^(1/3)*x+(x^3+2)^(2/3))/x^2)+1/3*3^(1/6)*arctan(1/9*3^(1/2) *(2*3^(2/3)*(x^3+2)^(1/3)+3*x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (59) = 118\).
Time = 1.36 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.97 \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx=\frac {1}{27} \cdot 3^{\frac {2}{3}} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} {\left (x^{3} + 2\right )}^{\frac {1}{3}} x^{2} - 2 \cdot 3^{\frac {2}{3}} {\left (x^{3} - 1\right )} - 9 \, {\left (x^{3} + 2\right )}^{\frac {2}{3}} x}{x^{3} - 1}\right ) - \frac {1}{54} \cdot 3^{\frac {2}{3}} \log \left (\frac {3 \cdot 3^{\frac {2}{3}} {\left (7 \, x^{4} + 2 \, x\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} + 3^{\frac {1}{3}} {\left (31 \, x^{6} + 46 \, x^{3} + 4\right )} + 9 \, {\left (5 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) - \frac {1}{9} \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (12 \cdot 3^{\frac {2}{3}} {\left (7 \, x^{7} - 5 \, x^{4} - 2 \, x\right )} {\left (x^{3} + 2\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 402 \, x^{6} + 192 \, x^{3} + 8\right )} - 18 \, {\left (31 \, x^{8} + 46 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} + 2\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (251 \, x^{9} + 462 \, x^{6} + 24 \, x^{3} - 8\right )}}\right ) \] Input:
integrate(1/(x^3-1)/(x^3+2)^(1/3),x, algorithm="fricas")
Output:
1/27*3^(2/3)*log((9*3^(1/3)*(x^3 + 2)^(1/3)*x^2 - 2*3^(2/3)*(x^3 - 1) - 9* (x^3 + 2)^(2/3)*x)/(x^3 - 1)) - 1/54*3^(2/3)*log((3*3^(2/3)*(7*x^4 + 2*x)* (x^3 + 2)^(2/3) + 3^(1/3)*(31*x^6 + 46*x^3 + 4) + 9*(5*x^5 + 4*x^2)*(x^3 + 2)^(1/3))/(x^6 - 2*x^3 + 1)) - 1/9*3^(1/6)*arctan(1/3*3^(1/6)*(12*3^(2/3) *(7*x^7 - 5*x^4 - 2*x)*(x^3 + 2)^(2/3) - 3^(1/3)*(127*x^9 + 402*x^6 + 192* x^3 + 8) - 18*(31*x^8 + 46*x^5 + 4*x^2)*(x^3 + 2)^(1/3))/(251*x^9 + 462*x^ 6 + 24*x^3 - 8))
\[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx=\int \frac {1}{\left (x - 1\right ) \sqrt [3]{x^{3} + 2} \left (x^{2} + x + 1\right )}\, dx \] Input:
integrate(1/(x**3-1)/(x**3+2)**(1/3),x)
Output:
Integral(1/((x - 1)*(x**3 + 2)**(1/3)*(x**2 + x + 1)), x)
\[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}} \,d x } \] Input:
integrate(1/(x^3-1)/(x^3+2)^(1/3),x, algorithm="maxima")
Output:
integrate(1/((x^3 + 2)^(1/3)*(x^3 - 1)), x)
\[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx=\int { \frac {1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}} \,d x } \] Input:
integrate(1/(x^3-1)/(x^3+2)^(1/3),x, algorithm="giac")
Output:
integrate(1/((x^3 + 2)^(1/3)*(x^3 - 1)), x)
Timed out. \[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx=\int \frac {1}{\left (x^3-1\right )\,{\left (x^3+2\right )}^{1/3}} \,d x \] Input:
int(1/((x^3 - 1)*(x^3 + 2)^(1/3)),x)
Output:
int(1/((x^3 - 1)*(x^3 + 2)^(1/3)), x)
\[ \int \frac {1}{\left (-1+x^3\right ) \sqrt [3]{2+x^3}} \, dx=\int \frac {1}{\left (x^{3}+2\right )^{\frac {1}{3}} x^{3}-\left (x^{3}+2\right )^{\frac {1}{3}}}d x \] Input:
int(1/(x^3-1)/(x^3+2)^(1/3),x)
Output:
int(1/((x**3 + 2)**(1/3)*x**3 - (x**3 + 2)**(1/3)),x)